9

9. [Maximum mark: 16]

In this question all values of $x$ and $t$ are in radians.

Consider the function $f(x) = 6\cos(\pi x)$.

(a) (i) Write down the amplitude of the graph of $f$.

(ii) Find the period of $f$. [3]

Consider a second function $g(x) = -8\sin(\pi x)$.

The sum of these functions can be expressed in the form $f(x) + g(x) = a\cos(b(x - c))$, where $a, b, c > 0$.

(b) By considering the graph of $y = f(x) + g(x)$, determine
(i) the value of $a$;

(ii) the value of $b$;

(iii) the smallest possible value of $c$. [4]

A car is travelling along a straight residential street with speed bumps placed at regular intervals on the road to encourage safer driving. The car travels at a minimum velocity when passing over speed bumps and reaches a maximum velocity in between speed bumps.

Its velocity, in ms$^{-1}$, can be modelled by the function $v(t) = -4.5\cos\left(\frac{\pi}{19}(t - 4)\right) + 10.5$, where $t$ is measured in seconds.

(c) Find the time at which the car first reaches its maximum velocity. [1]

(d) Find the number of speed bumps the car passes over in the first two minutes of motion. [1]

(e) (i) Find $v'(t)$.

(ii) Hence, or otherwise, write down the maximum acceleration of the car. [4]

(f) Find the distance, in metres, between consecutive speed bumps. [3]